# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021094
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## A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout

 1 School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China 2 Technology and Equipment of Rail Transit Operation and, Maintenance Key Laboratory of Sichuan Province, Chengdu 610031, China 3 Avic Chengdu Aircraft Industrial (Group)Co., Ltd, Chengdu 610031, China 4 School of Marketing, University of Southern Mississippi, Hattiesburg, MS 39406, USA

* Corresponding author: Chao Meng

Received  August 2020 Revised  March 2021 Early access May 2021

Fund Project: The first author is supported by China Postdoctoral Science Foundation (No.2020M673279), National Natural Science Foundation of China (NSFC) (No.51675450), Sichuan Science and Technology Program (No.2020JDTD0012) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No.18YJC630255)

Based on the rapid development of e-commerce, major promotional events and holidays can lead to explosive growth in market demand and place significant pressure on distribution systems. In this study, we considered a distribution system in which products are first transported to transfer satellites from a central depot and then delivered to customers from the transfer satellites. We modeled this distribution problem as a two-echelon vehicle routing problem with demand blowout (2E-VRPDB). We adopt a time-division distribution strategy to address massive delivery demand in two phases by offering incentives to customers who accept flexible delivery dates. We propose a hybrid fireworks algorithm (HFWA) to solve the 2E-VRPDB model. This model fuses an optimal cutting algorithm with an improved fireworks algorithm. To demonstrate the effectiveness and efficiency of the proposed HFWA, we conducted comparative analysis on a genetic algorithm and ant colony algorithm using a VRP example set. Finally, we applied the proposed model and HFWA to solve a distribution problem for the Jingdong Mall in Chengdu, China. The computational results demonstrate that the proposed approach can effectively reduce logistical costs and maintain a high service level.

Citation: Min Zhang, Guowen Xiong, Shanshan Bao, Chao Meng. A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021094
##### References:

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##### References:
Schematic diagram of the 2E-VRPDB
Initial second level solution schematic diagram
Explosive operation Ⅰ (2-opt)
Explosive operation II
3-opt operation
Flow chart of the HFWA
Optimal results of three algorithms
Jindong distribution of self-pickup points in Jinniu District
List of parameters and descriptions
 Sets and Parameters Description D Set of depots, $D=\{d_0\}$ S Set of satellites, $S=\{s_1$, $s _2 $$,…, s_{ns} }, and the total number is ns C Set of customers, C=\{c_1 , c_2 , …, c_{nc} \}, and the total number is nc G Set of first-level delivery vehicles, G=\{g _1 , g_2 , …, g _{ng} \}, and the total number is ng H Set of second-level delivery vehicles, H=\{h _1 , h _2 , …, h_{nh} \}, and the total number is nh M A large enough number T _0 Working hours per day d _{ij} The distance of the (i, j) edge q _i Demand of customer c _i cap _1 The capacity of the first-level vehicle cap _2 The capacity of the second-level vehicle t _c The deadline of customer c b _1 Compensation per unit cargo for accepting flexible delivery b _2 Delay cost per delivery a _1 Fixed cost of the first-level delivery vehicle per delivery a _2 Fixed cost of the second-level delivery vehicle per delivery c _g Unit distance cost of the first-level delivery vehicle per delivery c _h Unit distance cost of the second-level delivery vehicle per delivery c _1 The labor cost of the first-level delivery vehicle per delivery c _2 The labor cost of the second -level delivery per delivery f _c If customer c chooses flexible delivery, fc =1; otherwise, fc =0 T _s Time required to complete standard delivery  Sets and Parameters Description D Set of depots, D=\{d_0\} S Set of satellites, S=\{s_1 , s _2$$,…, s_{ns}$}, and the total number is $ns$ C Set of customers, $C=\{c_1$, $c_2$, …, $c_{nc} \}$, and the total number is $nc$ G Set of first-level delivery vehicles, $G=\{g _1$, $g_2$, …, $g _{ng} \}$, and the total number is $ng$ H Set of second-level delivery vehicles, $H=\{h _1$, $h _2$, …, $h_{nh} \}$, and the total number is $nh$ M A large enough number T$_0$ Working hours per day d$_{ij}$ The distance of the $(i, j)$ edge q$_i$ Demand of customer c$_i$ cap$_1$ The capacity of the first-level vehicle cap$_2$ The capacity of the second-level vehicle t$_c$ The deadline of customer c b$_1$ Compensation per unit cargo for accepting flexible delivery b$_2$ Delay cost per delivery a$_1$ Fixed cost of the first-level delivery vehicle per delivery a$_2$ Fixed cost of the second-level delivery vehicle per delivery c$_g$ Unit distance cost of the first-level delivery vehicle per delivery c$_h$ Unit distance cost of the second-level delivery vehicle per delivery c$_1$ The labor cost of the first-level delivery vehicle per delivery c$_2$ The labor cost of the second -level delivery per delivery f$_c$ If customer c chooses flexible delivery, fc =1; otherwise, fc =0 T$_s$ Time required to complete standard delivery
List of variables and descriptions
 Variables Description x$_{ijg}$ First-level distribution vehicle g travels the $(i, j)$ edge, $x _{ijg} $$\in \{0,1\}; decision variable y _{ijg} Second -level distribution vehicle h travels the (i, j) edge, y _{ijg} \in \{0,1\}; decision variable z _{cs} Customer c cargo comes from satellite s, z _{cs} \in \{0,1\}; decision variable w _{sg} The actual load of first level vehicle g to satellite s; decision variable l _s The total demand of satellite s t _{sg} Time of first-level vehicle g arriving at satellite s t _{ch} Time of arrival of second-level vehicle h to satellite s t _1 The longest time for the first-level vehicle to complete the distribution task time _c The actual delivery time of the customer c dtime _c The delay time for customer c U1 _{ig} Restrict the occurrence of sub-tour in the first-level vehicles U2 _{ih} Restrict the occurrence of sub-tour in the second-level vehicles u _{sg} Intermediate variable, no actual meaning u _{cp} Intermediate variable, no actual meaning  Variables Description x _{ijg} First-level distribution vehicle g travels the (i, j) edge, x _{ijg}$$ \in \{0,1\}$; decision variable y$_{ijg}$ Second -level distribution vehicle h travels the $(i, j)$ edge, y$_{ijg}$ $\in \{0,1\}$; decision variable z$_{cs}$ Customer c cargo comes from satellite s, z$_{cs}$ $\in \{0,1\}$; decision variable w$_{sg}$ The actual load of first level vehicle g to satellite s; decision variable l$_s$ The total demand of satellite s t$_{sg}$ Time of first-level vehicle g arriving at satellite s t$_{ch}$ Time of arrival of second-level vehicle h to satellite s t$_1$ The longest time for the first-level vehicle to complete the distribution task time$_c$ The actual delivery time of the customer c dtime$_c$ The delay time for customer c U1$_{ig}$ Restrict the occurrence of sub-tour in the first-level vehicles U2$_{ih}$ Restrict the occurrence of sub-tour in the second-level vehicles u$_{sg}$ Intermediate variable, no actual meaning u$_{cp}$ Intermediate variable, no actual meaning
Parameter settings
 Algorithm Parameter Value HFWA Fireworks population size 5 The number of explosion sparks 2 Upper limit of the number of explosion sparks 50 Variation spark number 2 Number of iterations 1000 ACO Number of ants 50 Pheromone heuristic factor 1 Fitness heuristic factor 9 Pheromone volatile factor 0.1 Constant coefficient 1 Number of iterations 1000 GA Population size 50 Cross factor 0.8 Mutation factor 0.2 Number of iterations 1000
 Algorithm Parameter Value HFWA Fireworks population size 5 The number of explosion sparks 2 Upper limit of the number of explosion sparks 50 Variation spark number 2 Number of iterations 1000 ACO Number of ants 50 Pheromone heuristic factor 1 Fitness heuristic factor 9 Pheromone volatile factor 0.1 Constant coefficient 1 Number of iterations 1000 GA Population size 50 Cross factor 0.8 Mutation factor 0.2 Number of iterations 1000
The optimization results of ACO algorithm
 No Standard test n ACO Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 422.93 422.93 50.7 1.40% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 387.84 50.5 0.75% 384.96 3 Set2a_E-n22-k4-s9-19 22 479.05 484.18 49.1 1.80% 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 377.56 49.1 1.63% 371.5 5 Set2a_E-n33-k4-s1-9 33 753.75 768.28 74.9 3.23% 730.16 6 Set2a_E-n33-k4-s2-13 33 761.76 776.57 75 6.60% 714.63 7 Set2a_E-n33-k4-s3-17 33 745.38 759.23 74.9 5.36% 707.48 8 Set2a_E-n33-k4-s7-25 33 790.55 804.92 75 4.45% 756.85 9 Set2a_E-n33-k4-s14-22 33 797.87 802.72 75.7 2.42% 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 618.69 637.24 124.9 16.57% 530.76 11 Set2b_E-n51-k5-s2-17 51 665.23 684.06 120.8 11.34% 597.49 12 Set2b_E-n51-k5-s4-46 51 613.78 627.29 120.2 15.64% 530.76
 No Standard test n ACO Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 422.93 422.93 50.7 1.40% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 387.84 50.5 0.75% 384.96 3 Set2a_E-n22-k4-s9-19 22 479.05 484.18 49.1 1.80% 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 377.56 49.1 1.63% 371.5 5 Set2a_E-n33-k4-s1-9 33 753.75 768.28 74.9 3.23% 730.16 6 Set2a_E-n33-k4-s2-13 33 761.76 776.57 75 6.60% 714.63 7 Set2a_E-n33-k4-s3-17 33 745.38 759.23 74.9 5.36% 707.48 8 Set2a_E-n33-k4-s7-25 33 790.55 804.92 75 4.45% 756.85 9 Set2a_E-n33-k4-s14-22 33 797.87 802.72 75.7 2.42% 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 618.69 637.24 124.9 16.57% 530.76 11 Set2b_E-n51-k5-s2-17 51 665.23 684.06 120.8 11.34% 597.49 12 Set2b_E-n51-k5-s4-46 51 613.78 627.29 120.2 15.64% 530.76
The optimization results of GA algorithm
 No. Standard test n GA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 438.04 472.4 0.00% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 399.37 468.9 0.75 % 384.96 3 Set2a_E-n22-k4-s9-19 22 475.62 492.41 474.9 1.07 % 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 383.11 472.1 1.63 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 764.25 502.2 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 725.04 747.5 484.6 1.46 % 714.63 7 Set2a_E-n33-k4-s3-17 33 732.37 760.82 488.5 3.52 % 707.48 8 Set2a_E-n33-k4-s7-25 33 763.58 790.26 491.8 0.89 % 756.85 9 Set2a_E-n33-k4-s14-22 33 782.04 792.21 510.7 0.38 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 599.66 631.44 860.2 12.98 % 530.76 11 Set2b_E-n51-k5-s2-17 51 641.66 671.12 695.5 7.39 % 597.49 12 Set2b_E-n51-k5-s4-46 51 604.92 620.14 656.7 13.97 % 530.76
 No. Standard test n GA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 438.04 472.4 0.00% 417.07 2 Set2a_E-n22-k4-s8-14 22 387.84 399.37 468.9 0.75 % 384.96 3 Set2a_E-n22-k4-s9-19 22 475.62 492.41 474.9 1.07 % 470.6 4 Set2a_E-n22-k4-s10-14 22 377.56 383.11 472.1 1.63 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 764.25 502.2 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 725.04 747.5 484.6 1.46 % 714.63 7 Set2a_E-n33-k4-s3-17 33 732.37 760.82 488.5 3.52 % 707.48 8 Set2a_E-n33-k4-s7-25 33 763.58 790.26 491.8 0.89 % 756.85 9 Set2a_E-n33-k4-s14-22 33 782.04 792.21 510.7 0.38 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 599.66 631.44 860.2 12.98 % 530.76 11 Set2b_E-n51-k5-s2-17 51 641.66 671.12 695.5 7.39 % 597.49 12 Set2b_E-n51-k5-s4-46 51 604.92 620.14 656.7 13.97 % 530.76
The optimization results of HFWA algorithm
 No. Standard test n HFWA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 417.07 103.7 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 386.69 106.2 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 472.84 107.1 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 376.35 104.2 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 734.76 188.9 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 714.63 724.6 191.1 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 707.48 712.08 192.1 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 765.18 192.1 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 779.05 781.95 194.7 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 530.76 557.82 593.6 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 597.49 622.8 490 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 530.76 549.47 499.2 0.00 % 530.76
 No. Standard test n HFWA Optimal solution (km) Optimal (km) Average (km) Time (s) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 417.07 103.7 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 386.69 106.2 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 472.84 107.1 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 376.35 104.2 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 730.16 734.76 188.9 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 714.63 724.6 191.1 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 707.48 712.08 192.1 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 765.18 192.1 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 779.05 781.95 194.7 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 530.76 557.82 593.6 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 597.49 622.8 490 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 530.76 549.47 499.2 0.00 % 530.76
The results of existing literature
 No. Standard test n [18] and [11] Optimal solution (km) Optimal (km) GAP (%) Optimal (km) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 0.00 % 417.07 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 0.00 % 384.96 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 0.00 % 470.6 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 0.00 % 371.5 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 743.22 1.79 % 730.16 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 710.48 -0.58 % 714.63 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 - - 707.48 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 0.00 % 756.85 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 - - 779.05 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 577.16 8.74 % 530.76 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 - - 597.49 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 - - 530.76 0.00 % 530.76
 No. Standard test n [18] and [11] Optimal solution (km) Optimal (km) GAP (%) Optimal (km) GAP (%) 1 Set2a_E-n22-k4-s6-17 22 417.07 0.00 % 417.07 0.00 % 417.07 2 Set2a_E-n22-k4-s8-14 22 384.96 0.00 % 384.96 0.00 % 384.96 3 Set2a_E-n22-k4-s9-19 22 470.6 0.00 % 470.6 0.00 % 470.6 4 Set2a_E-n22-k4-s10-14 22 371.5 0.00 % 371.5 0.00 % 371.5 5 Set2a_E-n33-k4-s1-9 33 743.22 1.79 % 730.16 0.00 % 730.16 6 Set2a_E-n33-k4-s2-13 33 710.48 -0.58 % 714.63 0.00 % 714.63 7 Set2a_E-n33-k4-s3-17 33 - - 707.48 0.00 % 707.48 8 Set2a_E-n33-k4-s7-25 33 756.85 0.00 % 756.85 0.00 % 756.85 9 Set2a_E-n33-k4-s14-22 33 - - 779.05 0.00 % 779.05 10 Set2b_E-n51-k5-s2-4-17-46 51 577.16 8.74 % 530.76 0.00 % 530.76 11 Set2b_E-n51-k5-s2-17 51 - - 597.49 0.00 % 597.49 12 Set2b_E-n51-k5-s4-46 51 - - 530.76 0.00 % 530.76
Distribution network node coordinates
 Node X Y Node X Y Node X Y D 20639 18019 16 14533 5098 34 13728 8485 S1 807 16768 17 13623 8242 35 12730 4622 S2 33084 19137 18 13573 3064 36 12968 4261 1 11066 5223 19 15874 7803 37 12611 3804 2 10481 7350 20 11212 6558 38 15604 5195 3 16356 5406 21 13396 3457 39 16340 4248 4 14197 6453 22 11813 9258 40 15342 6701 5 9591 5209 23 15606 5339 41 12902 3362 6 12664 5236 24 17577 5196 42 12149 5210 7 10772 5910 25 10496 5801 43 13221 5054 8 15321 5178 26 17472 3701 44 13451 7415 9 15239 5209 27 13839 7873 45 15543 7984 10 13556 7147 28 16555 8635 46 13025 4248 11 16660 4104 29 9347 6362 47 17460 4241 12 12438 3987 30 9547 8857 48 10895 4818 13 13850 6882 31 17942 3752 49 13704 10362 14 15196 8050 32 11042 5921 50 12929 4844 15 12864 4804 33 11387 5026
 Node X Y Node X Y Node X Y D 20639 18019 16 14533 5098 34 13728 8485 S1 807 16768 17 13623 8242 35 12730 4622 S2 33084 19137 18 13573 3064 36 12968 4261 1 11066 5223 19 15874 7803 37 12611 3804 2 10481 7350 20 11212 6558 38 15604 5195 3 16356 5406 21 13396 3457 39 16340 4248 4 14197 6453 22 11813 9258 40 15342 6701 5 9591 5209 23 15606 5339 41 12902 3362 6 12664 5236 24 17577 5196 42 12149 5210 7 10772 5910 25 10496 5801 43 13221 5054 8 15321 5178 26 17472 3701 44 13451 7415 9 15239 5209 27 13839 7873 45 15543 7984 10 13556 7147 28 16555 8635 46 13025 4248 11 16660 4104 29 9347 6362 47 17460 4241 12 12438 3987 30 9547 8857 48 10895 4818 13 13850 6882 31 17942 3752 49 13704 10362 14 15196 8050 32 11042 5921 50 12929 4844 15 12864 4804 33 11387 5026
Demand and delivery time of customer points
 Node Demand (packages) Time (days) Node Demand (packages) Time (days) Node Demand (packages) Time (days) 1 91 3 18 46 3 35 302 6 2 224 3 19 49 3 36 94 6 3 215 3 20 85 3 37 249 6 4 53 6 21 277 6 38 248 3 5 39 6 22 84 6 39 125 3 6 164 6 23 268 6 40 130 3 7 316 6 24 80 3 41 25 3 8 112 3 25 306 3 42 75 6 9 192 3 26 115 3 43 175 6 10 74 3 27 65 3 44 256 6 11 247 6 28 83 6 45 307 6 12 84 6 29 203 6 46 42 3 13 166 6 30 156 6 47 186 3 14 230 3 31 116 6 48 100 6 15 293 3 32 273 3 49 57 6 16 316 6 33 192 3 50 110 6 17 180 6 34 181 3
 Node Demand (packages) Time (days) Node Demand (packages) Time (days) Node Demand (packages) Time (days) 1 91 3 18 46 3 35 302 6 2 224 3 19 49 3 36 94 6 3 215 3 20 85 3 37 249 6 4 53 6 21 277 6 38 248 3 5 39 6 22 84 6 39 125 3 6 164 6 23 268 6 40 130 3 7 316 6 24 80 3 41 25 3 8 112 3 25 306 3 42 75 6 9 192 3 26 115 3 43 175 6 10 74 3 27 65 3 44 256 6 11 247 6 28 83 6 45 307 6 12 84 6 29 203 6 46 42 3 13 166 6 30 156 6 47 186 3 14 230 3 31 116 6 48 100 6 15 293 3 32 273 3 49 57 6 16 316 6 33 192 3 50 110 6 17 180 6 34 181 3
Computational results for the regular delivery model
 Level Vehicle NO. Standard delivery vehicle route First-level vehicle delivery 1S D-S1-D 2S D-S1-D 3S D-S1-D 4S D-S1-D Second-level vehicle delivery 1 S1-21-18-41-S1 2 S1-37-12-S1 3 S1-35-6-S1 4 S1-42-33-48-1-S1 5 S1-32-20-S1 6 S1-25-5-S1 7 S1-29-2-S1 8 S1-30-22-49-34-S1 9 S1-17-27-S1 10 S1-44-10-13-S1 11 S1-4-40-14-S1 12 S1-45-19-28-S1 13 S1-24-47-31-26-S1 14 S1-11-39-S1 15 S1-3-23-S1 16 S1-38-8-S1 17 S1-7-S1 18 S1-9-S1 19 S1-16-43-S1 20 S1-50-15-36-S1 21 S1-46-S1 Delivery time (days) 7 Delay cost (yuan) 32240 Compensation cost (yuan) 0 Total cost (yuan) 2159128
 Level Vehicle NO. Standard delivery vehicle route First-level vehicle delivery 1S D-S1-D 2S D-S1-D 3S D-S1-D 4S D-S1-D Second-level vehicle delivery 1 S1-21-18-41-S1 2 S1-37-12-S1 3 S1-35-6-S1 4 S1-42-33-48-1-S1 5 S1-32-20-S1 6 S1-25-5-S1 7 S1-29-2-S1 8 S1-30-22-49-34-S1 9 S1-17-27-S1 10 S1-44-10-13-S1 11 S1-4-40-14-S1 12 S1-45-19-28-S1 13 S1-24-47-31-26-S1 14 S1-11-39-S1 15 S1-3-23-S1 16 S1-38-8-S1 17 S1-7-S1 18 S1-9-S1 19 S1-16-43-S1 20 S1-50-15-36-S1 21 S1-46-S1 Delivery time (days) 7 Delay cost (yuan) 32240 Compensation cost (yuan) 0 Total cost (yuan) 2159128
Computational results for the TDD model
 Delivery method Level Vehicle NO. TDD vehicle route Standard delivery First-level vehicle delivery 1S D-S1-D 2S D-S1-D Second-level vehicle delivery 1 S1-26-47-24-8-9-S1 2 S1-38-19-20-S1 3 S1-32-1-S1 4 S1-2-25-S1 5 S1-33-41-18-S1 6 S1-46-15-S1 7 S1-10-27-S1 8 S1-14-34-S1 9 S1-40-3-S1 10 S1-39-S1 Flexible delivery First-level vehicle delivery 1S* D-S1-D 2S* D-S1-D 3S* D-S1-D Second-level vehicle delivery 11 S1-23-13-S1 12 S1-17-44-S1 13 S1-45-28-49-S1 14 S1-22-31-11-S1 15 S1-16-43-S1 16 S1-6-37-12-S1 17 S1-21-36-50-S1 18 S1-42-35-48-S1 19 S1-7-5-S1 20 S1-29-30-S1 Delivery time (days) 6 Delay cost (yuan) 0 Compensation cost (yuan) 2239.5 Total cost (yuan) 1937381
 Delivery method Level Vehicle NO. TDD vehicle route Standard delivery First-level vehicle delivery 1S D-S1-D 2S D-S1-D Second-level vehicle delivery 1 S1-26-47-24-8-9-S1 2 S1-38-19-20-S1 3 S1-32-1-S1 4 S1-2-25-S1 5 S1-33-41-18-S1 6 S1-46-15-S1 7 S1-10-27-S1 8 S1-14-34-S1 9 S1-40-3-S1 10 S1-39-S1 Flexible delivery First-level vehicle delivery 1S* D-S1-D 2S* D-S1-D 3S* D-S1-D Second-level vehicle delivery 11 S1-23-13-S1 12 S1-17-44-S1 13 S1-45-28-49-S1 14 S1-22-31-11-S1 15 S1-16-43-S1 16 S1-6-37-12-S1 17 S1-21-36-50-S1 18 S1-42-35-48-S1 19 S1-7-5-S1 20 S1-29-30-S1 Delivery time (days) 6 Delay cost (yuan) 0 Compensation cost (yuan) 2239.5 Total cost (yuan) 1937381
Delivery time sensitivity analysis
 Standard Penalty Standard TDD second- Penalty Compensation Total cost delivery (yuan) delivery level arrival (yuan) cost of delivery time (day) cost (yuan) time (day) (yuan) (yuan) 2 40300 1881455 4 6250 2239.5 1783100 5 3750 2239.5 1781518 6 1250 2239.5 1779718 3 32240 1868263 4 5000 2239.5 1782724 5 2500 2239.5 1779725 6 0 2239.5 1775722 4 24180 1862604 5 2500 2239.5 1779704 6 0 2239.5 1777541 7 0 2239.5 1777541
 Standard Penalty Standard TDD second- Penalty Compensation Total cost delivery (yuan) delivery level arrival (yuan) cost of delivery time (day) cost (yuan) time (day) (yuan) (yuan) 2 40300 1881455 4 6250 2239.5 1783100 5 3750 2239.5 1781518 6 1250 2239.5 1779718 3 32240 1868263 4 5000 2239.5 1782724 5 2500 2239.5 1779725 6 0 2239.5 1775722 4 24180 1862604 5 2500 2239.5 1779704 6 0 2239.5 1777541 7 0 2239.5 1777541
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